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  "body": "\n## Statement\n\nLet $R$ be a commutative ring.\nLet $A$ be an $m \\times n$ matrix and $B$ an $n \\times m$ matrix over $R$.\nThen $\\tr(AB) = \\tr(BA)$, where $\\tr$ is the [[trace|0XC1RDBY7A]].\n\n## Remarks\n\nBoth sides equal $\\sum_{i=1}^{m} \\sum_{j=1}^{n} a_{ij} b_{ji}$.\nThe matrices $AB$ and $BA$ can have different sizes, and the identity still holds.\nA consequence is $\\tr(P^{-1} A P) = \\tr A$, so similar matrices have the same trace.\nFor three matrices, only cyclic permutations are allowed in general: $\\tr(ABC) = \\tr(BCA)$, but $\\tr(ABC)$ can differ from $\\tr(BAC)$.\n\n## Explanations\n\n### L1 human\n\nAlthough $AB$ and $BA$ are usually different matrices, they have the same trace.\nEach diagonal sum adds the same products $a_{ij} b_{ji}$, only in another order.\nThis fact makes the trace a property of a linear map, not of the basis that we choose.\n",
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